Problem: Simplify; express your answer in exponential form. Assume $z\neq 0, n\neq 0$. $\dfrac{{(z^{-5}n^{3})^{2}}}{{z^{-2}n^{-1}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(z^{-5}n^{3})^{2} = (z^{-5})^{2}(n^{3})^{2}}$ On the left, we have ${z^{-5}}$ to the exponent ${2}$ . Now ${-5 \times 2 = -10}$ , so ${(z^{-5})^{2} = z^{-10}}$ Apply the ideas above to simplify the equation. $\dfrac{{(z^{-5}n^{3})^{2}}}{{z^{-2}n^{-1}}} = \dfrac{{z^{-10}n^{6}}}{{z^{-2}n^{-1}}}$ Break up the equation by variable and simplify. $\dfrac{{z^{-10}n^{6}}}{{z^{-2}n^{-1}}} = \dfrac{{z^{-10}}}{{z^{-2}}} \cdot \dfrac{{n^{6}}}{{n^{-1}}} = z^{{-10} - {(-2)}} \cdot n^{{6} - {(-1)}} = z^{-8}n^{7}$